Simulated Nanoscale Peeling Process of Monolayer Graphene Sheet - Effect of Edge Structure and Lifting Position

at the AFM tip/graphite interface has a wider space to be observed directly by ex. Transmission Electron Microscopy (TEM). This paper indicates the possibility of a direct observation of the stick-slip motion of the graphene sheet, that’s to say, the elementary process of the atomic-scale friction or superlubricity which occurs at the tip/graphite surface interface. The Stone Age, the Bronze Age, the Iron Age... Every global epoch in the history of the mankind is characterized by materials used in it. In 2004 a new era in material science was opened: the era of graphene or, more generally, of two-dimensional materials. Graphene is the strongest and the most stretchable known material, it has the record thermal conductivity and the very high mobility of charge carriers. It demonstrates many interesting fundamental physical effects and promises a lot of applications, among which are conductive ink, terahertz transistors, ultrafast photodetectors and bendable touch screens. In 2010 Andre Geim and Konstantin Novoselov were awarded the Nobel Prize in Physics "for groundbreaking experiments regarding the two-dimensional material graphene". The two volumes Physics and Applications of Graphene - Experiments and Physics and Applications of Graphene - Theory contain a collection of research articles reporting on different aspects of experimental and theoretical studies of this new material.


Introduction
Adhesion and peeling phenomena play important roles in connecting two objects regardless of whether they are inorganic, organic, or biological materials, which contributes to building up microscopic devices.The carbon nanostructures such as, carbon nanotube (CNT) and graphene have recently attracted great interests as the components of the electronic, magnetic, and optical devices.We have so far studied the peeling mechanics of the carbon nanotube (CNT) adsorbed onto the graphite surface both theoretically [1][2][3] and experimentally [4,5].It is clarified that the transition from the line-to the point-contact between the CNT and the graphite surface occurs during the peeling process [1][2][3][4][5].The CNT on the sub-microscale has the same size as the spatulae of the microscopic hairs aligned on the gecko foot [6,7].Therefore the study of the peeling process of the nanoscale objects such as, CNT is useful for not only developing the gecko-foot-mimic adhesives [8] but also understanding the elementary process of adhesion.
On the other hand, since the success of its experimental isolation [9], the potential of various applications of the graphene such as, the components of the electronic devices [10,11] has been discussed by many researchers.There is also a possibility that its adhesion with the substrate is applied to the adhesive tape at nanoscale.Therefore the peeling mechanics of the graphene sheet is very important, which can be regarded as the elementary process of the macroscopic sticky tape such as, the gecko-foot-mimic adhesives [6][7][8] or that of the microscopic extension of the crack in the fracture process.In our preliminary experiments, we have already succeeded in peeling the multilayered graphene plate with a thickness of several µm by using atomicforce microscopy tip [12].Here the two-component epoxy resin adhesive is used to bond the graphene plate to the AFM tip.Here the standard Si 3 N 4 tip for the contact AFM experiment is used.The junction formed between the AFM tip and the graphene should be mechanically rigid enough to measure the elasticity of the graphene sheet during the peeling process.The two-component epoxy resin adhesive satisfies the above condition.If the thickness of the peeled graphene plate is reduced, the comparison between the present simulation and the experiment will become possible.
Therefore, in this paper, ahead of experiment, we have theoretically reported the nanoscale peeling behaviors of the monolayer graphene sheet based on the molecular mechanics simulation [13,14].The peeling force curve exhibits the nanoscale change of the graphene shape from the surface to the line contact.The center position and the left edge are chosen as the lifting position.In Section 3, the peeling of the monolayer graphene sheet with the armchair edge for lifting the center position is discussed.In Sections 4 and 5, the peeling of the monolayer graphene sheet with the armchairand zigzag-edge for lifting the edge position is discussed, respectively.

Model and Method of Simulation
In the simulation, a rectangular-shaped monolayer graphene sheet with each side of 38 Å × 20 − 21 Å, comprised of 310 carbon atoms, is peeled from the rigid rectangular graphene sheet (which is called, the "graphite surface," hereafter) with each side of 164 − 165 Å × 58 Å, comprised of 3536 carbon atoms [Figure 1(a)].First, both the above graphene sheets are separately optimized by minimizing the covalent bonding energy described by the Tersoff potential energy [15], V cov , using the Polak-Rebiere-type conjugate gradient (CG) method [16].Here the convergence criterion is set so that the maximum of absolute value of all the forces acting on the movable atoms becomes lower than 10 −5 eV/ Å. Next, the graphene sheet is put and adsorbed onto the graphite surface, so that the AB stacking registry between the graphene sheet and the graphite surface is satisfied as shown in Figures 1(b) and 1(c).Here the green-colored six-membered ring at the center position or the outermost left edge of the graphene sheet is assumed to be attached to the AFM tip apex (Figure 1(a)), and then it is gradually moved upward along the z direction, parallel to the [0001] axis, by 0.1 Å.For each lifting position of the graphene sheet, z, the total energy V total = V cov + V vdW is minimized using the CG method, where V vdW is the nonbonding vdW interaction described by the modified Lennard-Jones (LJ) potential energy [17,18], acting between the graphene sheet and the graphite surface.Thus the optimized positions of the movable carbon atoms of the graphene sheet, (x, y, z), the vertical peeling force F z , and the lateral sliding forces F x and F y , acting on the lifting center, are calculated during the peeling process.In this paper, the graphene sheets with armchair-(Figure 1(b)) and zigzag-edges (Figure 1(c)) are discussed.

Center-Lifting Case of Armchair-Edge Graphene
When the six-membered ring located at the center position of the monolayer graphene sheet is lifted, the graphene sheet exhibits the characteristic transition of its shape during the peeling process within the x − z plane as illustrated in Figures 2(A surface.The vertical force F z is zero (Figure 3(a)).Just after the beginning of the peeling (Figure 2(B): z = 2.0 Å), the attractive interaction force takes the minimum value, −3.1 eV/ Å (Figure 3(B)).Once the line contacts are formed between the free edges (outermost arrays) of the peeled graphene sheet and the graphite surface, they clearly slide on the graphite surface as indicated by a circle in Figures 2(G) → 2(H) → 2(I) with a rapid increase of the bending of the graphene sheet.Within x-y plane, the right outermost array of the graphene sheet slides nearly straightforward along −x direction, not so sensitive to the lattice structure of the surface as illustrated in Figures 6(a  Thus the vertical peeling force F z exhibits the characteristic shape as shown in Figure 3, which reflects the transition from the surface to the line contact between the graphene sheet and the graphite surface.On the other hand, the lateral sliding force F x is zero due to the structural symmetry of the system.However, the lateral sliding force F y shows a finite value with an oscillation whose period and amplitude decrease as z increases.This oscillation of Figure 7 reflects the trajectory of the graphene edges illustrated in Figure 6 at the graphene-substrate interface during the peeling process.The maximum lateral force F y 0.1 eV/ Å which is only about 3% of the absolute value of the maximum adhesion force

Edge-Lifting Case of
Armchair-Edge Graphene       behavior of the F z curve decreases from 3.7 Å to 2.5 Å as shown in Figure 10(a) as the peeling proceeds.The lattice spacing of the graphite surface, 2.5 Å, appears in the peeling force curve particularly for the stick-slip region.

Edge-Lifting Case of Zigzag-Edge Graphene
Recently it has been reported that the edge structure of the graphene sheet plays quite an important role in electronic, magnetic, and optical properties of graphene, which can be also expected to give influences on the mechanical properties such as, the peeling process.Therefore, in this section, the peeling process of the graphene sheet with zigzag edge is discussed.In the simulation, the model obtained by rotating Figure 1(b) by 30 • is used (Figure 1(c)), and the left zigzag edge is lifted to simulate the peeling process, while the right free edge is zigzag type.As a result, the nanoscale peeling process within the x − z plane and the global shape of the force curve (Figure 12) is similar to that of Figures 8 and 9, respectively.The qualitative tendency of the decrease of the period and amplitude of the force curve (Figure 12(b)I-IV) is similar to that for Figure 9(a)I−VII.However, the details of the atomic-scale mechanics of the zigzag edge are clearly different from those of the armchair edge as follows.
During the surface contact, the graphene sheet first takes zigzag (Figures 13(b During the line contact, the difference between the armchair-and zigzag-type edge is enhanced.Figure 14(a  the hollow site or near the carbon bond, the graphene sheet bends toward the y direction to decrease the total interaction energy (Figures 14(b)2-3 and 6-7).Thus, in the case of the zigzag-type edge, collective motion of the single carbon "atom" on the free edge nearly dominates the graphene mechanics together with its deformation.On the other hand, for the armchair-type edge, collective motion of the single carbon "bond" is dominant.

Discussions and Conclusions
In this work molecular mechanics study of the nanoscale peeling of the monolayer graphene sheet has been performed.The peeling force curve clearly exhibits the change of the graphene shape from the surface-to the line-contact.In Section 3, the peeling of the monolayer graphene sheet with the armchair edge for lifting the center position is discussed.It is noted that the maximum lateral sliding force  3.1 eV/ Å.This small sliding force F y is derived from the superlubricity at the interface between the graphene sheet and the graphite surface [13] and atomic-scale wear [19].There is a possibility that such anisotropy between the vertical force F z and the lateral sliding force F y can be applied to the adhesives, which can be strongly adhered to the substrate but can easily slide on it.Our AFM measurement exhibits that the maximum pull-off force is about several hundreds of nN, which is clearly much larger than the binding force, 3 eV/ Å 4.8 nN, assumed in our simulation.
In Sections 4 and 5, the peeling of the monolayer graphene sheet with the armchair-and zigzag-edge for lifting the edge position is discussed, respectively.The atomicscale sliding motion of the monolayer graphene sheet during the peeling process is found.For the graphene sheet with armchair edge, the transition from the continuous to the stick-slip motion of the graphene sheet is found, which can be explained as follows.The peeling process induces the increase of the peeled area of the graphene sheet and the decrease of the surface contact area.Considering that the peeled area of the graphene sheet acts as an effective spring as shown in Figure 15, the increase of the peeled area makes the effective spring softer, and the decrease of the surface contact area decreases the energy barrier to slide the graphene sheet.Finally the peeling process induces the transition from the continuous to the stick-slip sliding motion of the graphene sheet, together with the decrease of the period and amplitude of the z − F z curve.An important point is that the period of the peeling force curve for the armchair-edge graphene for the surface contact region corresponds to the lattice spacing of the graphite surface along [1230] direction, 2.5 Å.On the other hand, for the zigzag-edge graphene, the period becomes the lattice spacing along [1010] direction, 4.4 Å.This means the sliding length of the graphene sheet along x direction becomes nearly equal to the peeled length along z direction.The zigzag structures of the peeling force curve with the same period of about several Å have been also observed by our preliminary experiments using the multilayered graphene, which will be reported elsewhere [12].Of course, if the number of the peeled graphene sheets is reduced, the direct comparison between the present simulation and the experiment will become possible.As a result, the center-lifting case requires the largest attractive peeling force, −3.1 eV/ Å, in order to peel the graphene sheet as shown in Figure 16(a).On the other hand, the edge-lifting case requires only −0.74 eV/ Å, about 20% of that for the center-lifting case as shown in Figures 16(b) and 16(c).The edge structures give little influences on the basic features of the force curve.However, the sliding direction and the edge structure clearly give marked influences on the surface-and line-contact regions, respectively.
Another important point is that the behavior of the lateral force curve (F x (z)) is qualitatively the same as that of the vertical force curve (F z (z)) during the surface contact as shown in Figure 9(b).Therefore, it can be said that the peeling force curve, F z (z), directly reflects the atomicscale friction force, F x (z), which decreases to 0.019 eV/ Å 30 pN for z = 27.8Å (Figure 9(b)).This ultralow friction force, F x , is derived from the superlubricity at the interface between the graphene sheet and the graphite surface [19][20][21].Furthermore, effect of the edge structure on the peeling process is clarified by comparison of the free edge between the armchair-and zigzag-types.As mentioned above, the atomic-scale structure of the force curve during the surface contact reflects the lattice spacing of the graphite surface.So the minimum period of the atomic-scale structure of the force curve can tell us the atomic-scale lattice orientation and structure of the free edge of graphene.Such information can be used for the control of the electronic properties of the graphene sheet adsorbed onto the substrate.Therefore, this paper indicates the possibility of the identification of the lattice orientation and the edge structure of the graphene sheet.
In this paper, we discussed the importance of the dynamics of the free edge during the peeling process.On the other hand, we also found the importance of the shape of the graphene sheet.Additional simulated model and results are shown in Figure 17.As shown in Figure 17(a), rectangular graphene sheet whose aspect ratio is different from that of the graphene sheet of Figure 1(b) is used.The basic shape of the vertical force curve, Figure 17(b), is similar to that of Figure 3.However, the armchair-type edge is peeled first for Figure 10(a), although the zigzag-type edge is peeled first for Figure 1(b) as discussed in Section III.This means that the shape of the graphene sheet plays an important role for deciding which edge is peeled first.Effect of the graphene shape on the peeling process will be discussed in detail somewhere in the near future.
Lastly it should be noted that the peeling process discussed in this paper is closely related to the atomic-scale wear of the graphite and the graphene tip formation in the friction force microscopy [22].When the tip is pushed onto the surface for less than the critical tip height, the outermost graphene layer is attached to the FFM tip, which results in the formation of the graphene tip.In that case, the graphene sheet takes the surface contact with the second layer graphene, and it takes the two-dimensional stick-slip motion.However, it is difficult to observe directly the stickslip motion during the scan process, due to the very small gap between the FFM tip and the graphite surface.On the other hand, if the peeling process is used, it can be expected that the contact at the AFM tip/graphite interface has a wider space to be observed directly by ex Transmission Electron Microscopy (TEM).This paper indicates the possibility of a direct observation of the stick-slip motion of the graphene sheet, that is to say, the elementary process of the atomic-scale friction or superlubricity which occurs at the tip/graphite surface interface.

Figure 1 :
Figure 1: (a) The schematic illustration of the model of the monolayer graphene sheet physically adsorbed onto the rigid graphite surface used in the simulation.The green-colored sixmembered ring at the center position or left edge of the graphene sheet is assumed to be adsorbed onto the atomic force microscopy tip apex indicated by broken lines, and it is moved upward along the z (or [0001]) direction, by z = 0.1 Å.Initial AB stacking registry of the red-colored graphene sheet with (b) armchair and (c) zigzag edge adsorbed onto the blue-colored graphite surface within the x − y plane.

Figure 2 :
Figure 2: The transition of the shape of the monolayer graphene sheet during the peeling process from A to J within the x-z plane.The red-colored graphene sheet and blue-colored graphite surface are shown.The displacement of the lifting center position from the initial position, z( Å), is indicated on the upper-right positions of each picture.

Figure 3 :
Figure 3: The vertical force, F z , acting on the center six-membered ring, plotted as a function of the lifting displacement z.The positions A−J correspond to those of Figure 2.
) and 6(b), which show the trajectories of the two carbon atoms on the right outermost array illustrated in Figure1(b).The sliding of the outermost arrays during G and I appears much more clearly than that during A and G.During H and I, the decrease of F z (Figures3(H) and 3(I)) can be explained by the decrease of the repulsive force acting on the carbon atoms on the left and right edges of the graphene sheet as shown in Figure5, that is to say, the relative increase of the effect of the attractive interaction force.

Figure 4 :
Figure 4: The atomic structures of the graphene sheet just before and after the discrete change, B → C, D → E, and I → J.The regions surrounded by dotted ellipses show the partial peeled areas.

4. 1 .
Figures 9(a)A−9(a)J, the vertical force acting on the lifting edge F z plotted as a function of the edge height z.

Figure 5 :
Figure 5: The averaged forces acting on one atom on the left and right outermost arrays (red-colored) and those on the left and right second arrays (blue-colored), as a function of the displacement of the lifting center position from the initial position, z( Å).

Figure 6 :
Figure 6: The trajectory of the two carbon atoms on the right free edge (outermost array) indicated by white circles in Figure 1(b).(a) The whole trajectory A → I, and (b) the part of the trajectory A → E, including the discrete jumps, B → C and D → E, are indicated.White circles mean carbon atoms of the graphite surface.The indices A−I correspond to those in Figure 2.

Figure 7 :Figure 8 :
Figure 7: The lateral force, F y , acting on the center six-membered ring, plotted as a function of the displacement z.The positions A−J correspond to those of Figure 2.

Figure 9 :
Figure 9: (a) The vertical force, F z , acting on the lifting edge, plotted as a function of the lifting edge height z for the graphene with armchair-type free edge.The indices A−J correspond to those of Figure 8.(b) The red-colored averaged force per one atom acting on the outermost array, and the blue-colored one acting on the second array, as a function of the lifting edge height z( Å).The indices E, F, H, and I correspond to those of Figures 8 and 9(a).
Figure 9: (a) The vertical force, F z , acting on the lifting edge, plotted as a function of the lifting edge height z for the graphene with armchair-type free edge.The indices A−J correspond to those of Figure 8.(b) The red-colored averaged force per one atom acting on the outermost array, and the blue-colored one acting on the second array, as a function of the lifting edge height z( Å).The indices E, F, H, and I correspond to those of Figures 8 and 9(a).

Figure 10 :
Figure 10: (a) Enlargement of part of the z − F z curve (Figure 9(a)) corresponding to the continuous and stick-slip process during the surface contact.(b) The trajectories of the two carbon atoms on the free edge from 1 to 5 indicated in (a).(c) The trajectories of the two carbon atoms on the free edge from 6 to 10 indicated in (a).

Figure 9 (Figure 11 :
Figure 11: (a) Enlargement of part of the z − F z curve (Figure 9(a)) corresponding to the nearly straight stick-slip region during the line contact.(b) The trajectories of the two carbon atoms on the free edge from 1 to 3 indicated in (a).

2 → 3 )
corresponds to the stick-slip sliding motion of the graphene sheet (Figures11(b)1 → 2 → 3).Here the free edge of the graphene sheet slides with nearly the straight stick-slip motions.One of the carbon atoms on the free edge passes over the carbon-carbon bonds as shown in Figures11(b)1 and 3.

Figure 12 :
Figure 12: The vertical force, F z , acting on the lifting edge, plotted as a function of the lifting edge height z for the graphene with zigzag-type free edge.
)1−6) and then straight stickslip motions (Figures 13(c)7−11), passing over the nearest neighboring AB-stacking site along [1010] direction.It is noted to avoid AA-stacking registry, the graphene sheet takes zigzag slip toward the nearest neighboring AB-stacking site as shown in Figure 13(b)1 → 2, although it takes straight slip as shown in Figure 13(c)7 → 8.The minimum period of the force curve of 4.4 Å (Figure 13(a)IV) reflects the lattice period of the graphite surface along the [1010] direction, while 2.5 Å for the armchair-type edge (Figure 10(a)VII) reflects that along the [1230] direction.Thus the edge structure gives the marked effects on the atomic-scale dynamics depending on the lattice orientation of the surface.
) reflects the zigzag stick-slip motion of the graphene sheet (Figures 14(b)1−9) unlike nearly the straight stick-slip motion (Figures 10(b)1−3).Important point of the linecontact sliding is that each carbon atom on the free edge takes stick-slip motion between the nearest neighboring sixmembered rings.When each atom is located on the hollow site of the six-membered ring, the graphene sheet does not deform along the y direction (Figures 14(b)1, 4-5, and 8-9).However, when each atom is located a little far from

Figure 13 :
Figure 13: (a) Enlargement of part of the z−F z curve (Figure 12(b)) corresponding to the zigzag and straight stick-slip process during the surface contact.(b) The trajectories of the two carbon atoms on the free edge from 1 to 6 indicated in (a).(c) The trajectories of the two carbon atoms on the free edge from 7 to 11 indicated in (a).

Figure 14 :
Figure 14: (a) Enlargement of part of the z−F z curve (Figure 12(b)) corresponding to the zigzag stick-slip process during the line contact.(b) The trajectories of the two carbon atoms on the free edge from 1 to 9 indicated in (a).

Figure 15 :
Figure 15: (a) Schematic illustration of the increase of the peeled area and the decrease of the surface contact area from C (z = 6.1 Å) to D (z = 18.3 Å) for the graphene sheet with armchair-type free edge.(b) −F z and F x plotted as a function of the edge height z, show qualitatively the same behavior to each other for the graphene sheet with armchair-type free edge.

F y 0. 1
eV/ Å is only about 3% of the absolute value of the maximum adhesion force |F z |

Figure 16 :
Figure 16: Comparison of the vertical forces, F z , among (a) centerlifting case of the armchair-edge graphene sheet, (b) edge-lifting case of the armchair-edge graphene sheet, and (c) edge-lifting case of the zigzag-edge graphene sheet.

Figure 17 :
Figure 17: (a) The model of the red-colored monolayer graphene sheet physically adsorbed onto the blue-colored rigid graphite surface within the x-y plane.The green-colored six-membered ring at the center position is moved upward along the z (or [0001]) direction, by z = 0.1 Å.Initial AB stacking registry of the redcolored graphene sheet with the blue-colored graphite surface is assumed.(b) The vertical force, F z , acting on the center sixmembered ring, plotted as a function of the lifting displacement z.